Question: $ 0.\overline{47} \div 0.\overline{19} = {?} $
Explanation: First convert the repeating decimals to fractions. $\begin{align*} 100x &= 47.4747...\\ x &= 0.4747...\end{align*} $ $\begin{align*} 99x &= 47 \\ x &= \dfrac{47}{99}\end{align*} $ $\begin{align*} 100y &= 19.1919...\\ y &= 0.1919...\end{align*} $ $\begin{align*} 99y &= 19 \\ y &= \dfrac{19}{99}\end{align*} $ So, the problem becomes: $ \dfrac{47}{99} \div \dfrac{19}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{47}{99} \times \dfrac{99}{19} = {?} $ $ \phantom{\dfrac{47}{99} \times \dfrac{19}{99}} = \dfrac{47 \times 99}{99 \times 19} $ $ \phantom{\dfrac{47}{99} \times \dfrac{19}{99}} = \dfrac{47 \times \cancel{99}} {\cancel{99} \times 19} $ $ \phantom{\dfrac{47}{99} \times \dfrac{19}{99}} = \dfrac{47}{19} $